As is known in the art, the compressor outlet temperature of a turbocharger cannot exceed the capability of the material of the compressor outlet housing under all of turbocharged engine operating conditions; not only for the turbocharged engine at sea level conditions, but also when the turbocharger is operating at altitude to ensure adequate operating margins. If any of the mechanical or thermal loading limits are exceeded, boost pressure or fueling is decreased and recalculate the new turbocharger operating points are recalculated to find satisfactory conditions. The conventional method to calculate the compressor outlet temperature uses compressor efficiency to obtain relatively accurate results. However, this method cannot be extended to a two-stage turbo charger because the efficiency map of a two-stage compressor cannot be derived directly and further it is relatively difficult to maintain adequate accuracy without extensive experiments.
As is known, using the thermal second law analysis, for a compressor, assuming that the compression process is isentropic, the following relation between the temperature and pressure at the inlet (Tc—in, pc—in) and at the outlet (Tc,is, pc—out) the compressor can be derived:
                              (                                    T                              c                ,                is                                                    T                              c                ⁢                _                ⁢                in                                              )                =                              (                                          p                                  c                  ⁢                  _                  ⁢                  out                                                            p                                  c                  ⁢                  _                  ⁢                  in                                                      )                                              γ              -              1                        γ                                              (        1        )            However, due to enthalpy losses across the compressor the compression process is not isentropic in reality. Therefore, the compressor isentropic efficiency, ηc, is introduced which relates the theoretical temperature rise (leading to Tc,is) to the actual (resulting in Tc—out) where:
                              η          c                =                                            T                              c                ,                is                                      -                          T                              c                ⁢                _                ⁢                in                                                                        T                              c                ⁢                _                ⁢                out                                      -                          T                              c                ⁢                _                ⁢                in                                                                        (        2        )            Substituting this into (1) yields the expression:
                              η          c                =                                                            (                                                      p                                          c                      ⁢                      _                      ⁢                      out                                                                            p                                          c                      ⁢                      _                      ⁢                      in                                                                      )                                                              γ                  -                  1                                γ                                      -            1                                                              T                                  c                  ⁢                  _                  ⁢                  out                                                            T                                  c                  ⁢                  _                  ⁢                  in                                                      -            1                                              (        3        )            where γ is the specific heat ratio. Then the temperature downstream of the compressor from (3):
                                          Π            T                    =                                                    T                                  c                  ⁢                  _                  ⁢                  out                                                            T                                  c                  ⁢                  _                  ⁢                  in                                                      =                          1              +                                                1                                      η                    c                                                  ⁢                                  (                                                            Π                      p                                                                        γ                          -                          1                                                γ                                                              -                    1                                    )                                                                    ⁢                                  ⁢        where        ⁢                                  ⁢                                            Π              p                        =                                          P                                  c                  ⁢                  _                  ⁢                  out                                                            P                                  c                  ⁢                  _                  ⁢                  in                                                              ;                ⁢                                  ⁢                              Π            T                    =                                                    T                                  c                  ⁢                  _                  ⁢                  out                                                            T                                  c                  ⁢                  _                  ⁢                  in                                                      .                                              (        4        )            
The compressor efficiency, ηc, is the ratio of isentropic rise to the actual temperature rise across the compressor, and is used to compensate for the losses caused by other physical effects which are difficult to model. Since the compressor efficiency, ηc, varies little along the steady state operating point, it is typically modeled with a map, called the compressor efficiency map of lines of constant efficiency, ηc, shown in FIG. 1, which is a function of the pressure ratio, Πp, of compressor and reduced air mass flow. Thus, for a measured mass airflow and a ratio Πp of measured output pressure to measured input pressure, the compressor efficiency, ηc, can be determined from the map in FIG. 1. Having the compressor efficiency, ηc, from the map and a measured input temperature, Tc—in, the output temperature, Tc—out, can be calculated from equation (4). However, with the “island-like” efficiency lines, as shown in FIG. 1, it would require significant effort to extend the range of available experimental data on a flow bench or engine cell, rather than trying to predict or extrapolate the behavior outside of the given range, even some points inside of the given range. Further, applying this process to a second, cascaded compressor (i.e., a two-stage turbocharger) would require additional temperature and pressure sensors.